Difference between function and polynomials

There's a subtle difference between them. A polynomial is not a priori a function, but can be used as a function. By definition, a polynomial is just a bunch of numbers strung up next to symbols of the form $x^n$. It is a formal object. For example, let's say $$p(x)=a_nx^n + a_{n-1}x^{n-1} + \cdots a_1 x + a_0.$$

Then to say $p(x)$ is zero as a polynomial, by definition means that all $a_i$ are zero. It turns out that if you were to consider $p(x)$ as a function, saying $p(a) = 0$ for all real numbers $a$, also implies that all $a_i$ must be zero, but now this is not by definition and needs a proof:

Suppose that $p(a)=0$ for all $a\in \mathbb{R}$. We want to show $p(x)=0$ as a polynomial, so we want to show all $a_i=0$. Now if $p(x)$ were a constant polynomial, then it would have to be the zero constant, so we would be done. If $p(x)$ were not constant, then $p(a)=0$ implies $x-a$ is a a factor of $p(x)$. But $p(x)$ can not have infinitely many factors, so that's a contradiction. Therefore, $p(x)=0$ as a polynomial.

Similarly, if we say $p(x)=q(x)$ as polynomials, we mean all their coefficients are equal. But if we say $p(x)=q(x)$ as functions, that implies they are equal as polynomials.

So it turns out that the two notions are compatible, but often when we say $p(x)$ is a polynomial, we really just mean a bunch of numbers written next to powers of $x$.

There is certainly a formal difference, as explained in Prometheus' answer. But let's ignore that for a while, and think about how general functions differ from those defined by polynomials. Maybe this is what the OP had in mind, anyway.

Also, let's assume that we're talking about real-valued functions defined on some bounded interval. If the function is continuous, then the Weierstrass approximation theorem tells us that it can be approximated uniformly to within any desired tolerance by a polynomial. So, in a sense, such a function is almost a polynomial. And, for numerical/computational purposes, it is a polynomial (or can be adequately represented as one).

So, looking at things this way, there is a fairly broad set of situations where polynomials and general functions are essentially the same thing. If you're working in more exotic/abstract settings, it's a different story, of course.